Problem 16 - Algebra 2

Question

Simplify each expression. \(\frac{30 b c}{12 b^{3}}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{5c}{2b^2} \).

Step 1: Identify Common Factors

First, we need to identify any common factors in the numerator and denominator. In this case, the expression is \( \frac{30bc}{12b^3} \). Both the numerator and the denominator contain the variable \( b \). Additionally, we notice that both 30 and 12 share a common factor of 6.

Step 2: Simplify the Coefficients

Divide the coefficients (numbers) by their greatest common divisor. Divide 30 by 6 to get 5, and divide 12 by 6 to get 2. Replace the coefficients in the expression: \( \frac{30bc}{12b^3} = \frac{5bc}{2b^3} \).

Step 3: Simplify the Variables

Now, simplify the expression by reducing the powers of \( b \). The numerator has \( b \) to the power of 1, and the denominator has \( b \) to the power of 3. Cancel one \( b \) from both the numerator and denominator to get \( \frac{c}{2b^2} \).

Step 4: Rewriting the Simplified Expression

Write the simplified expression in its final form: \( \frac{5c}{2b^2} \). This is the simplest form of the given expression after factoring and canceling common terms.

Key Concepts

Common Factors

To simplify a rational expression, we begin by identifying any common factors in the numerator and the denominator. Common factors are numbers or variables that appear in both parts of the fraction.

In our expression, \( \frac{30bc}{12b^3} \), both the numerator (30bc) and the denominator (12b³) share the variable \( b \). Finding common factors is like looking for patterns that are present in both parts, making the expression easier to simplify.

For instance, the numbers 30 and 12 each have 6 as a common factor. This means both numbers can be divided evenly by 6. Recognizing these common factors can significantly simplify our calculations and lead us to the most simplified form of the expression.

Greatest Common Divisor

An important step in simplifying any numerical part of a rational expression is to find the greatest common divisor (GCD). The GCD of two numbers is the largest number that divides both numbers evenly. This helps to reduce fractions to their simplest form.

In our exercise, the coefficients (numerical parts of the expression) are 30 and 12. To simplify these numbers, we determine the GCD of 30 and 12, which is 6.

• Divide 30 by 6 to get 5.
• Divide 12 by 6 to get 2.

After performing these divisions, the expression changes to \( \frac{5bc}{2b^3} \). This process is crucial before moving on to simplifying the variable parts.

Variable Cancellation

In the final step of simplifying our expression, we address the variable \( b \). The technique of variable cancellation focuses on reducing terms by removing the same variable from both the numerator and the denominator as much as possible.

Take a look at \( b \) in \( \frac{5bc}{2b^3} \). The numerator features \( b \) to the power of 1, while the denominator has \( b \) raised to the power of 3. We can cancel \( b \) from both parts of the fraction.

• Remove one \( b \) from the numerator and three \( b \)s from the denominator.
• This leaves us with \( \frac{c}{2b^2} \).

After removing the common variable terms, our expression is now in its simplest form: \( \frac{5c}{2b^2} \). This process ensures all possible simplifications are made, making your expression clear and concise.